Phase Separation explains a new class of selforganized spatial patterns in ecological system s
Q uan-X ing Liua b-1, A rjen D oelm anc, Vivi R o ttsch äferc, M onique d e Ja g er3, P ete r M. J. H erm an3, M ax R ietkerkd,
a n d Jo h an van d e K oppei3 0
d e p a r tm e n t of Spatial Ecology, Royal N etherlands Institute for Sea Research, 4400 AC Yerseke, The Netherlands; bAquatic Microbiology, Institute for
Biodiversity and Ecosystem Dynamics, University of Am sterdam, 1090 GE Am sterdam, The Netherlands; 'M athem atical Institute, Leiden University, 2300 RA
Leiden, The Netherlands; dD epartm ent of Environmental Sciences, Copernicus Institute, Utrecht University, 3508 TC Utrecht, The Netherlands; and
'C om m unity and Conservation Ecology Group, Centre for Ecological and Evolutionary Studies, University of Groningen, 9700 CC Groningen, The Netherlands
Edited by Simon A. Levin, Princeton University, Princeton, NJ, and approved June 10, 2013 (received for review December 20, 2012)
mussels | m athem atical model | spatial self-organization | animal
aggregation
activator-inhibitor principle, originally conceived by Turing
Thein 1952
(1), provides a potential theoretical mechanism for
the occurrence of regular patterns in biology (2-6) and chemistry
(7-9), although experimental evidence in particular for biological
systems has remained scarce (3, 4, 10). In the past decades, this
principle has been applied to a wide range of ecological systems,
including arid bush lands (11-15), mussel beds (16,17), and boreal
peat lands (18, 19). The principle, in which a local positive acti­
vating feedback interacts with large-scale inhibitory feedback to
generate spatial differentiation in growth, birth, mortality, respi­
ration, or decay, explains the spontaneous emergence of regular
spatial patterns in ecosystems even under near-homogeneous
starting conditions. Physical theory offers an alternative m echa­
nism for pattern formation, proposed by Cahn and Hilliard in
1958 (20). They identified that density-dependent rates of dis­
persal can lead to separation of a mixed fluid into two phases
that are separated in distinct spatial regions, subsequently lead­
ing to pattern formation. The principle of density-dependent
dispersal, switching between dispersion and aggregation as local
density increases, has become a central mathematical explana­
tion for phase separation in many fields (21) such as multiphase
fluid flow (22), mineral exsolution and growth (23), and bi­
ological applications (24-28). Although aggregation due to in­
dividual motion is a commonly observed phenom enon within
ecology, application of the principles of phase separation to ex­
plain pattern formation in ecological systems is absent both in
terms of theory and experiments (25, 26).
H ere, we apply the concept of phase separation to the for­
m ation of spatial patterns in the distribution of aggregating
mussels. On intertidal flats, establishing mussel beds exhibit
www.pnas.org/cgi/doi/10.1073/pnas.1222339110
spatial self-organization by forming a pattern of regularly spaced
clumps. By so doing, they balance optimal protection against pre­
dation with optimal access to food, as demonstrated in a field ex­
periment (29). This self-organization process has been attributed to
the dependence of the speed of movement on local mussel density
(29 ). Mussels move at high speed when they occur in low density and
decrease their speed of movement once they are included in small
clusters. However, when occurring in large and dense clusters, they
tend to move faster again, due to food shortage. Mussel pattern
formation is a fast process, giving rise to stable patterning within the
course of a few hours, and clearly is independent from birth or death
processes (Fig. 1 A andB ). Although mussel pattern formation at
centimeter scale was successfully reproduced by an empirical individual-based model (29), to date no satisfactory continuous model
has been reported that can identify the underlying principle in
a general theoretical context.
In this paper, we present the derivation and analysis of a partial
differential equation model based on an empirical description of
density-dependent movement in mussels, and demonstrate that it
is mathematically equivalent to the original model of phase sep­
aration by Cahn and Hilliard (20). We then compare the pre­
dictions of this model with observations from real mussel beds and
experiments with mussel pattern formation in the laboratory.
Results
Model Description. Mussel speed of movement was observed to
initially decrease with increasing mussel density, but to increase
when the density exceeded that typically observed in nature (Fig.
1C). The movement speed data were fitted to the following
equation:
V (M) = aM2 - b M + c,
[1]
with a = 2.30, ft = 2.19, and c = 0.62 (Fig. 1C, blue line). A lin e ar
model proved not significant (P = 0.449). The quadratic model
was overall significant (P < 0.001), where the coefficient for the
second-order term was highly significant (t = 5.717, P < 0.0001),
and the Akaike information criterion test showed that the qua­
dratic model was highly preferable over the linear one (see Table
SI for details). N ote that we used a quadratic equation not b e­
cause it provides the most optimal fit, but because it is the most
simple polynomial function.
A uthor contributions: Q.-X.L. an d J.v.d.K. designed research; Q.-X.L., A.D., V.R., and M.d.J.
perform ed research; Q.-X.L. analyzed d ata; and Q.-X.L., P.M.J.H., M.R., and J.v.d.K. w ro te
th e paper.
The au th o rs declare no conflict o f interest.
This article is a PNAS Direct Submission.
Freely available online th ro u g h th e PNAS o p en access option.
1To w hom co rresp o n d en c e should b e ad d ressed . E-mail: Iiuqx315@ gmail.com.
This article contains supporting inform ation online a t www.pnas.org/lookup/suppl/doi: 10.
1073/pnas. 1222339110/-/DCSupplemental.
PNAS | July 16,2013 | vol. 110 | no. 29 | 11905-11910
APPLIED
MATHEMATICS
The origin o f re g u la r sp a tia l p a tte rn s in ecological sy ste m s h a s
long fa scin a te d research ers. T u rin g 's a c tiv a to r-in h ib ito r principle
is c o n sid ere d th e cen tral p a rad ig m t o e xplain such p a tte rn s .
A ccording t o th is principle, local activ atio n c o m b in ed w ith longra n g e inhibition o f g ro w th a n d survival is an e sse n tia l p re re q u isite
fo r p a tte rn fo rm atio n . H ere, w e s h o w t h a t th e physical principle o f
p h a se se p a ra tio n , solely b a se d o n d e n s ity -d e p e n d e n t m o v em e n t
by o rg an ism s, re p re s e n ts an a lte rn a tiv e class o f self-o rg an ized
p a tte rn fo rm a tio n in ecology. Using e x p e rim e n ts w ith se lf-o rg a ­
nizing m ussel b e d s, w e d eriv e a n em pirical re la tio n b e tw e e n th e
sp e e d o f anim al m o v e m e n t a n d local anim al d e n sity . By incorpo­
ra tin g th is re la tio n in a p a rtial d iffere n tia l e q u a tio n , w e d e m o n ­
s tr a te t h a t th is m odel c o rre sp o n d s m ath e m a tic ally t o th e w ellk n o w n C ahn-H illiard e q u a tio n fo r p h a se se p a ra tio n in physics.
Finally, w e sh o w t h a t th e p re d ic te d p a tte rn s m atch th o s e fo u n d
b o th in field o b se rv a tio n s a n d in o u r e x p erim e n ts. O ur re su lts re ­
veal a principle fo r ecological se lf-o rg a n iza tio n , w h e re p h a se se p ­
a ra tio n ra th e r th a n activ atio n a n d inhibition p ro c esses driv es
sp a tia l p a tte rn fo rm atio n .
0.8r
9{m)
V(m)
1 0.6
i
X 0.4
0.2
m
0.0
0.0
0.2
0.4
0.6
0.8
Rescaled mussel density
1.0
v51
.
Fig. 1. Pattern form ation in mussels and statistical properties of th e density-dependent m ovem ent of mussels under experim ental laboratory conditions.
(A and B) Mussels th a t w ere laid o ut evenly under controlled conditions on a hom ogeneous substrate developed spatial patterns similar to "labyrinth-like"
after 24 h (images represent a surface of 60 cm x 80 cm). (C) Relation betw een m ovem ent speed and density within a series of mussels clusters (mussel density
is rescaled, w h ere 128 equals to 1). The blue line describes th e rescaled second-order polynomial fit with Eq. 1. The red line depicts th e effective diffusion g(m)
of mussels as a function of th e local densities according to th e diffusion-drift theory. The open circles show th e original experim ent data, and th e solid squares
represent th e average speed of each group. (D) The numerical simulation of Eq. 4 im plem ented with param eters fi = 1.89, Do = 1.0, and Ari =0.1, sim ulating th e
developm ent of spatial p atterns from a near-uniform initial state.
Based on this formulation, we now derive an equation for the
changes in local density M of a population of mussels, in a 2D
space. As the model describes pattern formation at timescales
shorter than 24 h, growth and mortality (as factors affecting local
mussel density) can be ignored. Local fluxes of mussels at any
specific location can therefore be described by the generic con­
servation equation:
dM
-= - V / .
[2 ]
Here, / is the net flux of mussels, and V = (A, dy) is the gradient
in two dimensions.
To derive the net flux/, we assume that mussel movement can
be described as a random, step-wise walk with a step size V that is
a function of mussel density, and a random, uncorrelated reor­
ientation. In the case of density-dependent movement, the net
flux arising from the local gradient in mussel density can be
expressed as follows (SI Text):
/,=
1
2t
v(v+M^~
V
dM
VAf,
[3]
where r is the turning rate, following Schnitzer (30) (equation
4.14 of ref. 30). The “drift” term M d V / d M accounts for the
effect of spatial variation in local mussel density on the spatial
flux of mussels. This term does not appear in the case of densityindependent movement, but its contribution is crucial when upscaling the density-dependent movement of individuals to the
population level.
Following earlier treatm ents of biological diffusion as a result
of individual movement (2), we complement this local diffusion
11906 I www.pnas.org/cgi/doi/10.1073/pnas.1222339110
term with a term that accounts for the long-distance movement
by including nonlocal diffusion as I,t¡ = V( k AM) with nonlocal
diffusion coefficient k. The nonlocal diffusion process has a rel­
atively low intensity, and hence param eter k is much smaller in
magnitude than the local movement coefficient in Eq. 3. W e can
now gather both fluxes into the total net flux rate in Eq. 2,
ƒ = / , +ƒ„/, to define the general rescaled conservation equation
as follows (SI Text):
dm
—- = D 0V[g(m) -A'iV(Am)].
ot
Here, g(m) =v(m) \v(m)
[4]
where v ( m ) = m 2 —ßm + 1
is a rescaled speed. D 0 is a rescaled diffusion coefficient that
describes the average mussel movement, and k\ is the rescaled
nonlocal diffusion coefficient. Rescaling at the basis of Eq. 4
is given by the following relations: g(m) = (m 2 - ßm + l)(3»z2 2ßm + l) with m = \Ja/cM, Do = Ç , k\=^4-, and ß = b/\/äc.
Here, ß captures the depression of diffusion at intermediate
densities in a single param eter. In this model, spatial patterns
develop once the inequalities ß > \/3 a n d ß < 2 are satisfied, lead­
ing to a negative effective diffusion (aggregation) g(m) at inter­
mediate mussel densities. Thus, if mussel movement is significantly
depressed at intermediate density, then effective diffusion g(m)
becomes negative, mussels aggregate, and patterns emerge. If
the depression of mussel movement speed at intermediate mussel
density is weak, then g(m) remains positive, and no aggregation
occurs at intermediate biomass (Fig. SI). U nder these conditions,
no patterns emerge. The fitted values for a, b, and c reveal that the
effective diffusion clearly can become negative (as ß = 1.8339), as
shown in Fig. 1C (red line). Eq. 4 predicts the formation of regular
p attern s (Fig. ID ), in close agreem ent w ith the p attern s as
Liu e t al.
observed in our experiments (Fig. 15). Using the precise param ­
eter setting obtained from our experiments, we are able to dem ­
onstrate that reduced mussel movement v(m) at intermediate
mussel density results in an effective diffusion g{m) that can
change sign, which leads to the observed formation of patterns.
A Physical Principle. We now show that Eq. 4 is mathematically
S
o
Si
ds
— = D V 2[P(s ) - ÂrAs] = HV [P'(s)Vs- kV( As) },
-
[5]
where P(s) typically has the form of the cubic s3 - 5. In S I Text,
we show that density-dependent functions of g(m) of Eq. 4 and
its corresponding expression P'(s) in Eq. 5 have the same m ath­
ematical shape (concave upward) with two zero solutions, p ro ­
vided that movement speed V (M) remains positive for all values
of M, which is inevitably valid for any animal. Hence, in a similar
way as described in the C ahn-H illiard equation, net aggregation
of mussels at interm ediate densities generates two phases, one
being the mussel clump, the other having such a low density that
it can be identified with open space, given the discrete nature of
the mussels. This occurs due to a decrease in movement speed at
interm ediate density, leading to net aggregation when g{m) < 0,
similar to what is predicted by the C ahn-H illiard equation (Fig.
S2). Hence, we find that pattern formation in mussel beds fol­
lows a process that is principally similar to phase separation,
triggered by a behavioral response of mussels to encounters
with conspecifics.
Com parison of Experim ental Results and M odel's Predictions. Eq. 4
yields a wide variety of spatial patterns with increasing mussel
density, which are in close agreement with the patterning ob­
served in the field (Fig. 2), as well as in laboratory experiments
(Fig. S3). Theoretical results dem onstrate that, with the specific
value of ß determ ined in our experiment, four kinds of spatial
patterns can emerge, depending on mussel density. W hen mussel
numbers are increased from a low value, a succession of patterns
develops from sparsely distributed dots (Fig. 2E) to a “labyrinth
p attern” (Fig. 2F) and a “gapped pattern” (Fig. 2G), and finally
the patterns weaken before disappearing (Fig. 2H). Note that the
theoretical results closely match the patterns observed in the
field (Fig. 2 A - D ) . Moreover, a similar succession of patterns has
been found under controlled experimental conditions (29) when
the number of mussels is increased (Fig. S3). The spatial cor­
relation function of the images obtained during the experiments
generally agrees with that of the patterns predicted by Eq. 4,
displaying a dam ped oscillation that is characteristic of regular
patterns (Fig. S4 and SI Text).
A similar agreement was found in the emergence and disap­
pearance of spatial patterns with respect to changing mussel
numbers when we com pared a numerical analysis with an ex­
perim ental bifurcation analysis. The mathematical simulation
predicts that the amplitude of the aggregative pattern (i.e., the
maximal density observed in the pattern) dramatically increases
with increasing overall mussel densities, but decreases again
when mussel density becomes high (Fig. 3A). Most significantly,
these predictions are qualitatively confirmed by our laboratory
experiments, as shown in Fig. 3B. W e observed an increase in the
amplitude when the num ber of mussels in the arena was low, but
a rapid decline of the amplitude with increasing overall mussel
numbers when mussel numbers were high, in agreement with the
general predictions of the model. It should be noted that,
Liu e t al.
(
■ -0
i
I
■ -0
■ -o
I
Pattern form ation of mussels in th e field and numerical results for
2D simulations w ith varying densities. (A -D ) Mussel patterns in th e field
varying respectively from isolated clumps, "open labyrinth," "gapped p a t­
tern ed " to a dense, near-hom ogeneous bed. (E-H) Changes in simulated
spatial patterning in response to changing overall density, closely follow th e
field observed patterns. The color bar shows values of th e dimensionless
density m of Eq. 4. Simulation param eters are th e same as for Fig. 1D ap art
from th e overall density of mussels.
Fig. 2.
although spatial homogeneity can easily be obtained in simulated
patterns, the discrete nature of living mussels precludes this in
our experiments, especially at low mussel density. Hence, the
agreem ent should only be sought in qualitative terms.
As a final test of equivalence to the C ahn-H illiard model, we
investigated whether pattern formation in mussels exhibits a
coarsening process referred to as the Lifshitz-Slyozov (LS) law
(21, 31). Here, the spatial scale of the patterns, f(t), grows in a
power-law m anner as f(t) ex tr, where the growth exponent y = 1/3
was found to be characteristic of the C ahn-H illiard equation
(31-33). O ur experimental results reveal that this scaling law also
holds remarkably well during early pattern formation in mussel
July 16, 2013
I vol. 110 | no. 29
ECOLOGY
S
I
APPLIED
MATHEMATICS
rj
21
equivalent to the well-known Cahn-H illiard equation for phase
separation in binary fluids (see SI Text for detailed discussion).
The original Cahn-H illiard equation describes the process by
which a mixed fluid spontaneously separates to form two pure
phases (20, 27). The C ahn-H illiard equation follows the general
mathematical structure:
Mussel density (cm 2)
Mussel density (cm 2)
Fig. 3. Bifurcation of th e am plitude of patterns as a function of mussel density as predicted by th e theoretical model (A) and found in th e experim ental
p atterns (ß). (A) P aram eter values are fi = 1.89, Do = 1.0, and /ci =0.1, apart from mussel density; letters indicate position on th e plot corresponding to th e four
snapshots E, F, G, and H in Fig. 2. The mussel density represents values of th e dimensionless density. (B) Laboratory m easurem ent of pattern ed am plitudes
w ith d ifferent densities on surface of 30 cm x 50 cm, w here th e num ber of mussels ranges from 100 to 1,400 individuals. Am plitude versus th e m ean density is
depicted as symbol lines with solid squares; th e red lines depict average density.
beds, where we found a scaling exponent very close to 1/3 during
the first 6 h of self-organization, independent of mussel density
(Fig. 4). Moreover, this behavior is independent of mussel den­
sity. However, the LS scaling law collapses at a later stage as the
mussels settle and attach to each other with byssus threads. The
theoretical model (Eq. 4) matches this result, displaying the
same scaling exponent as our experiments, of course without the
collapse of the scaling law, because the model does not take into
account other long-term biological processes.
Discussion
The results reported here establish a general principle for spatial
self-organization in ecological systems that is based on density-
10 °
Ti me, t (min)
Fig. 4. Scaling properties of th e coarsening processes. The relation betw een
spatial scale versus tem poral-increasing on th e pattern form ation in doublelogarithmic scale. The colored solid lines indicate th e experim ental data for
different mussel densities and th e theoretical simulation. The dashed lines fit
th e experim ental d ata with a pow er law ((t) ccP a t early stages. W e found
only a slight deviation from th e theoretically expected y = 1/3 grow th. No
dom inant w avelength em erges from th e spectral analysis for th e first
minutes of th e experim ent, and hence no d ata could be plotted. Note that,
as th e simulation starts w ith a very fine-grained random distribution, p a t­
tern developm ent takes longer in th e model.
11908 I www.pnas.org/cgi/doi/10.1073/pnas.1222339110
dependent movement rather than scale-dependent activatorinhibitor feedback. This principle is akin to the physical process
of phase separation, as described by the Cahn-H illiard equation
(20). Density-dependent movement has until now not been rec­
ognized as a general mechanism for pattern formation in ecology,
despite aggregation by individual movement being a commonly
described phenom enon in biology (28, 34—37). Recent theoreti­
cal studies highlight similar aggregative processes as a possible
mechanism behind pattern formation in microbial systems (26,
38, 39), insect migration (25), or passive movement as found in
stream invertebrates (40). Furtherm ore, studies on ants and
termites have shown that self-organization can result from indi­
viduals actively transporting particles, aggregating them onto
existing aggregations to form spatial structures ranging from
regularly spaced corps piles (41 ) to ant nests (42). Also, a number
of studies highlight that, beyond food availability (43), behavioral
aggregation in response to predator presence is an im portant
determ inant of the spatial distribution of birds (44). These
studies indicate there may be a wide potential for application of
the Cahn-H illiard framework of phase separation in ecology and
animal behavior that extends well beyond our mussel case study.
A fundamental difference exists between pattern formation as
predicted by Turing’s activator-inhibitor principle and that p re­
dicted by the C ahn-H illiard principle for phase separation.
Characteristic of Turing patterns is that a homogeneous “back­
ground state” becomes unstable with respect to small spatially
periodic perturbations: this so-called Turing instability is the
driving mechanism behind the generation of spatially periodic
Turing patterns. Moreover, the fixed wavelength of these p at­
terns is determ ined by this instability. In the Cahn-H illiard
equation, there is no such “unstable background state” that can
be seen as the core from which patterns grow. As we have seen
(Fig. 4), the Cahn-Hilliard equation, as well as our model, exhibits
a coarsening process: the wavelength slowly grows in time. Hence,
Cahn-Hilliard dynamics have the nature of being forced to in­
terpolate between two stable states, or phases, whereas a Turing
instability is “driven away from an unstable state.”
Strikingly, in mussels, both processes may occur at the same
time. Mussels aggregate because they experience lower mortality
due to dislodgement or predation in clumps (29). This explains
why, on the short term, they aggregate in a process that, as we
argue in this paper, can be described by Cahn and H illiard’s
model for phase separation. On the long term, however, they
settle and attach to other mussels using byssal threads, a process
th a t arrests p attern form ation, thereby disabling the coarsen­
ing n atu re of “p u re” C ah n -H illiard dynamics by a biological
Liu e t al.
Laboratory Setup and Mussel Sampling. The laboratory setup follow ed th a t of
a previous study by Van d e Koppei e t al. (29). Pattern form ation by mussels
was studied in th e laboratory w ithin a 130 cm x 90 cm x 27 cm polyester
container filled w ith seaw ater. Mussel sam ples w ere obtained from w ooden
w ave-breaker poles on th e beaches near Vlissingen, The N etherlands
(51.458713N, 3.531643E). They w ere kept in containers and fed live cultures
of Phaeodactylum tricornutum daily. In th e experim ents, mussels w ere laid
o u t evenly on a surface of eith er concrete tiles or a red PVC sheet. The
container w as illum inated using fluorescent lamps. Fresh, unfiltered sea­
w ater was supplied to th e container a t a rate of ~1 L/min.
Imaging Procedures and Mussels' Tracking. The m ovem ent of individual mussels
Pattern Amplitude Determination. The analysis of th e am plitude of th e mussel
p atterns w as based on tw o experim ental series. In th e first series, 450, 750,
1,200, and 1,850 mussels w ere evenly spread over a 60 cm x 80 cm red PVC
sheet. In th e second series, 100, 200, 400, 600, 1,100, and 1,400 mussels w ere
evenly spread over a 30 cm x 50 cm sheet. W e analyzed small-scale variation
in mussel density from th e im age recorded by th e webcam a fte r 24 h using
a moving w indow of 3 cm x 5 cm, in which th e mussels w ere counted. The
maximum density was used as th e am plitude o f th e p attern . Four typical
images are show n in Fig. S2.
Calculation of the Scale of the Patterns. The spatial scale of th e patterns w ere
obtained quantitatively by determ ining th e w avelength o f th e pattern s from
th e experim ental images. W e applied a 2D Fourier transform to obtain th e
pow er spectrum w ithin a square, moving w indow . Local w avelength was
identified for each w indow , and th e results w ere averaged for all windows.
This straightforw ard technique is suitable for identifying th e w avelength in
noisy images with irregular patterning (45).
Numerical Implementation. The continuum equation (Eq. 4) was sim ulated on
a HP Z800 w orkstation with an NVidia Tesla C1060 graphics processor. For
th e 2D spatial patterns, our com putation code w as im plem ented in th e
CUDA extension o f th e C language (www.nvidia.com/cuda). The spatial
fourth-order kernel is im plem ented in 2D space using th e numerical schemes
shown in Fig. S5. Spatial p atterns w ere obtained by Euler integration o f th e
finite-difference equation w ith discretization of th e diffusion (46). The
model's predictions w ere exam ined for different grid sizes and physical
lengths. W e ad o p ted periodic boundary conditions for th e rectangular
spatial grid. Starting conditions consisted of a hom ogeneous distribution of
mussels w ith a slight random perturbation. All results w ere obtained by
setting At = 0.001 and Ax = 0.15.
A CKNOW LEDGM ENTS. W e th a n k Franjo Weissing and Kurt E. A nderson fo r
was recorded by taking an im age every m inute using a Canon PowerShot D10,
which was positioned ab o u t 60 cm above th e w ater surface, and attached to
a laptop com puter. Each image contained th e entire experimental domain a t
a 3,000 x 4,000 pixels resolution. W e tested th e effect of increasing mussel
densities on m ovem ent speed. We set up a series of mussel clusters with 1 ,2 ,4 ,
6, 8, 16, 24, 32, 48, 64, 80, 104, and 128 mussels, respectively, on a red PVC
sheet to provide a contrast-rich surface for later analysis. The m ovem ent speed
of individual mussels w as obtained by measuring th e m ovem ent distance
along th e trajectories during 1 mín. All im age analysis and tracking programs
are developed in MATLAB (R2012a; The Mathworks).
critical com m ents on th e m anuscript, review er Nigel G oldenfeld fo r
suggestions, in particular for em phasizing th e im portance o f th e LifshitzSlyozov pow er law, and an anonym ous review er fo r constructive com ­
m ents. We also th a n k A niek van d e r Berg fo r assisting w ith th e labo rato ry
experim ents. This study w as financially su p p o rted by The N etherlands
O rganization o f Scientific Research th ro u g h th e N ational P rogram m e Sea
and Coastal Research Project W addenE ngine. The research of M.R. is
sup p o rted by th e European Research Area-Net on Complexity th ro u g h th e
project "Resilience and Interaction of Networks in Ecology an d Econom­
ics." The research o f J.v.d.K. is su p p o rte d by t h e M osselw ad Project,
fu n d e d by th e W ad d en fo n d s an d th e Dutch M inistry o f In frastru ctu re
a n d t h e E nvironm ent.
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PNAS | July 16, 2013 | vol. 110 | no. 29 | 11909
ECOLOGY
M aterials and M ethods
Field Photos of Mussel Patterns. Field photos of mussel patterns w ith different
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APPLIED
MATHEMATICS
mechanism th a t acts on interm ediate tim escales and has not
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Hence, on the long run, both demographic processes (16) and ag­
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real mussel beds.
Finally, our results dem onstrate that, to understand complexity
in ecological systems, we need to recognize the importance of
movement as a process that can create coherent spatial structure
in ecosystems, rather than just dissipate them. Unlike the growth/
mortality-based Turing mechanism, the movement-based C ahnHilliard mechanism has short timescales. It may thus allow for
fast adaptation and generate transient spatial structures in eco­
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even within the same ecosystems. How the interplay between
these two mechanisms affects the complexity and resilience of
natural ecosystems is an im portant topic for future research.
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